Typographical and other errors

Stephen G. Simpson

March 21, 2012

Typographical errors in the first edition

Here is a list of typographical errors in Subsystems of Second Order Arithmetic, Stephen G. Simpson, Springer-Verlag, 1999, XIV + 445 pages. It contains all errors which were discovered through 2006. All of these errors were corrected in the second edition which was published in 2009.

Proof of Theorem I.9.1, displayed formula, replace $\vert c-c_k\vert<2^n$ by
$\vert c-c_k\vert<2^{-n}$ .
Theorem I.10.3, in item 8, replace ``set of sentences'' by ``consistent set of sentences''.
Proof of Lemma III.2.5, the subscripts need to be repaired. The proof should be as follows:

We first consider the case of a finite product $\widehat A=\prod_{i=1}^m\widehat A_k$ . In this case, for each $j\in\mathbb{N}$ , let the smallest such that $m\cdot 2^{-l}\le 2^{-j}$ . Put $n_j=\prod_{k=1}^m(n_{l_jk}+1)-1$ and let $\langle x_{i j}: i\le n_j\rangle$ be an enumeration of $\prod_{k=1}^m\{x_{i l_j k}: i\le n_{l_j k}\}$ . Then $\langle\langle x_{i j}: i\le n_j\rangle: j\in\mathbb{N}\rangle$ attests to the compactness of $\widehat A$ .

In the case of a countably infinite product $\widehat A=\prod_{k\in\mathbb{N}}\widehat A_k$ , for each $j\in\mathbb{N}$ let smallest such that $(j+2)\cdot2^{-l}\le 2^{-j-1}$ . Put $n_j=\prod_{k=0}^{j+1} (n_{l_jk}+1)-1$ and let $\langle x_{i j}: i\le n_j\rangle$ be an enumeration of $\prod_{k=0}^{j+1}\{x_{i l_j k}: i\le n_{l_j k}\}$ . Again $\langle\langle x_{i j}: i\le n_j\rangle: j\in\mathbb{N}\rangle$ attests to the compactness of $\widehat A$ . This completes the proof of the lemma.
Proof of Theorem III.3.2, main paragraph, replace $h:K\to\mathbb{Q}$ by
$g:K\to\overline\mathbb{Q}$ , and replace by . Also, replace all three occurrences of by .
Proof of Theorem III.4.3, last line of third paragraph, replace
$\forall n (q_{2n}\ne0\to f(q_{2n+1}/q_{2n})=n)$
by
$\forall n (q_{2n}\ne0\to f(q_{2n+1}/q_{2n})=n)$ and $\forall n (q_{2n}=0\to q_{2n+1}=0)$ .
Proof of Theorem III.6.5, middle of page 120, replace $y_{ij}=p_{f(i)}z_{i,j+1}$ by $y_{ij}=p_{f(i)}y_{i,j+1}$ .
Proof of Theorem III.6.5, first line of last paragraph on page 120, replace $h_i:A\to D,i=1,2$ by $h_1,h_2:A\to D$ .
Proof of Lemma IV.1.4, end of proof, replace $f^{\textstyle{*}}(\sum^{j-1}_{i=0}f(i)+k)=1$ by
$f^{\textstyle{*}}(\sum^{j-1}_{i=0}g(i)+k)=1$ .
Exercise IV.2.10, replace ``closed set'' by ``closed set ''.
Definition V.4.5, replace A by .
Remark V.10.1, replace $\Sigma^0_1$ - $\mathsf{AC}_0$ by $\Sigma^1_1$ - $\mathsf{AC}_0$ .
Theorem VI.2.6, replace ``Over'' by ``over''.
Proof of Sublemma VI.3.3, in the definition of , replace $X_1\oplus X_1$ by $X_0\oplus X_1$ .
Definition VII.3.2.4, replace by .
Proof of Theorem VII.3.31, first line, replace T by .
Proof of Theorem VII.6.9, first paragraph, second last sentence, replace ``The'' by ``Then''.
Remark VIII.1.16, replace ``as do $\mathsf{ACA}_0$ and $\Sigma^1_1$ - $\mathsf{AC}_0$ '' by ``while $\mathsf{ACA}_0$ and $\Sigma^1_1$ - $\mathsf{AC}_0$ prove the same $\Pi^1_2$ sentences.''
Proof of Lemma VIII.2.16, first line of page 323, replace by .
Proof of Lemma IX.2.4, second line, replace $\vert M\vert\cup\mathcal{S}_{M'}$ by $\vert M\vert\cup\mathcal{S}_M$ .
Theorem X.2.9, items 5, 6, and 7, replace by $X^{\textstyle{*}}$ .
Section X.3, just before Definition X.3.1, replace ``Ramsey's theorem'' by ``Ramsey's theorem for exponent 3''.
Definition X.4.1, replace ``atomic formula'' by ``numerical term''.
Bibliography, item 268, replace ``borel'' by ``Borel''.
Index, atomic formula, replace ``2, 410'' by ``2''.
Index, formula, atomic, replace ``2, 410'' by ``2''.
Index, GKT basis theorem, replace ``325-326'' by ``325-326, 354-356''.
Index, numerical term, replace ``23'' by ``2, 23, 410''.
Index, term, numerical, replace ``2, 23'' by ``2, 23, 410''.
Index, add index entry, weak $\Sigma^1_1$ - $\mathsf{AC}_0$ , 342-347.
Index, universal $\Sigma^1_1$ , replace ``333, 356'' by ``252, 333, 356''.

Errors in the second edition

Here is a list of errors in Subsystems of Second Order Arithmetic, Second Edition, Stephen G. Simpson, Association for Symbolic Logic, Cambridge University Press, 2009, XVI + 444 pages. This list is current as of today, March 21, 2012.

The proof of Theorem IV.8.2 contains two typographical errors. In displayed equation (15), $<2^{-3(n+2)}$ should be $<-2^{-3(n+2)}$ . In line 7 from the bottom of the page, $>-2^{-3(n+2)}$ should be $>2^{-3(n+2)}$ .
David Madore has pointed out that part 18 of Lemma VII.3.7 is incorrect. The error is not typographical but mathematical. Namely, does not include the Axiom of Regularity, so for instance it is consistent with that there are proper-class-many sets such that , and of course all such sets are hereditarily finite.
There are two ways to repair this.
1. Preferred way: Move the Axiom of Regularity from Definition VII.3.8 (the axioms of $\mathsf{ATR}_0^{\mathrm{set}}$ ) to Definition VII.3.3 (the axioms of $\mathsf{B}_0^{\mathrm{set}}$ ).
2. Another way: Change part 6 of Definition VII.3.6 to read as follows:
  Trans $(u)\leftrightarrow u$ is transitive, i.e., $\forall x\,\forall y\,((x\in y\land y\in u)\rightarrow x\in u)\land\forall v\... ...t)\rightarrow\exists x\,(x\in v\land\forall y\,(y\in v\rightarrow y\notin x)))$ .
  Thus regularity is incorporated into the definitions of transitivity and hereditary finiteness.
Under either 1 or 2, part 7 of Definition VII.3.6 can be simplified by omitting the clause which begins with $\forall v$ .
In line 2 of the proof of Theorem VII.3.9, ``Then'' should be ``The''.
David Madore has pointed out another mathematical error. Namely, in the Notes for §VII.5 on page 293, the stated characterization of $\alpha^\Delta_{k+2}$ is correct for $k\ge1$ but incorrect for .
A correct characterization of $\alpha^\Delta_2$ reads as follows. $\alpha^\Delta_2=$ the least admissible ordinal which is the limit of smaller admissible ordinals.